EE8405 Chap.5 Deterministic Queues and M/M/1
Deterministic Queues and M/M/1
Deterministic queues are traced through known arrival and service events, while M/M/1 is analysed as a birth-death chain with random transitions. Comparing them isolates the effect of variability at equal mean rates.
The chapter derives the geometric stationary distribution, requires stability before formula use, reconciles queue size and delay through Little's formula, and shows the nonlinear congestion penalty near saturation.
What this chapter covers
- 01
Deterministic timelines
- 02
Kendall notation
- 03
Birth-death chains
- 04
M/M/1 stability
- 05
Mean queue size
- 06
Delay growth near saturation
Worked example · free
Audit an M/M/1 link
- 2Check utilization and stability.
- 3Calculate queue size and total delay.
- 3Separate waiting and reconcile through Little's formula.
Key terms
- Birth-death Chain
- A Markov chain whose transitions change the state by one step.
- Stability
- The condition under which a queue has finite long-run stationary measures.
Deterministic Queues and M/M/1 FAQ
Why can delay explode below full utilization?
Random arrival clusters create backlogs that require spare capacity to clear. As arrival rate approaches service rate, the clearing margin shrinks and mean delay grows nonlinearly.
How does D/D/1 differ from M/M/1?
At equal stable mean rates, deterministic spacing can eliminate waiting after transients, while exponential randomness produces overlap and a positive queue. The contrast isolates variability as the cause.
What is the first M/M/1 check?
Calculate rho as arrival rate divided by service rate. If rho is at least one, the standard infinite-buffer stationary formulas do not apply. Check this before substitution.
How can I verify an M/M/1 solution without repeating it?
Use independent identities. Total mean number should equal mean number waiting plus utilization, while total mean delay should equal mean waiting plus mean service time. Little's formula should connect each matched number and time through the same throughput. The stationary probabilities should sum to one and have a constant successive-state ratio equal to utilization.
These checks test boundaries, units and distributional reasoning rather than reusing the same algebra, so disagreement identifies the kind of error to inspect.
What should a deterministic queue timeline show?
Give every customer an arrival time, service-start time and departure time, then calculate waiting as service start minus arrival. State the convention for simultaneous events and verify that the server does not idle while work is waiting. The accumulated-work path should reveal whether a transient backlog clears under the stated mean rates.
Comparing this timeline with an M/M/1 result at equal rates isolates the effect of random clustering and service variation instead of confusing variability with additional demand.
Which endpoint checks belong in a final answer?
At light traffic, total delay should approach mean service time. As utilization approaches one from below, stationary queue size and delay should rise sharply. State both directions before calculating.
Exam move
Use endpoint checks: light traffic should approach mean service time, while delay should diverge as arrival rate approaches service rate from below. Start revision with a paired timeline. Give a deterministic server and an M/M/1 server the same arrival and service rates, then trace the first system event by event and solve the second from its stationary distribution.
The comparison makes variability visible without changing average demand or capacity. For every deterministic exercise, list arrival times, service-start times, departure times and waiting times in separate columns. State how simultaneous arrival and departure events are ordered. Check that nobody begins service before arriving and that a work-conserving server is never idle while work is waiting.
For M/M/1, write the physical assumptions before the formulas: Poisson arrivals, independent exponential services, one server, infinite waiting space and a work-conserving discipline. Calculate utilization before any stationary measure. If it is at least one, stop and explain why the usual long-run formulas do not apply. When it is below one, derive one primary quantity and obtain the others through identities.
Reconcile total number with number waiting plus utilization. Reconcile total delay with waiting plus mean service. Confirm the geometric state probabilities sum to one and that the ratio of consecutive probabilities equals utilization. Practise sensitivity by holding service rate fixed and moving arrival rate closer to it. Predict the direction first, then compute the new delay and spare service margin.
The nonlinear increase should be explained as reduced capacity for clearing random clusters, not merely described as a larger denominator effect. Finish each solution by naming whether time includes service and whether number includes the customer in service.
Use a limiting check at both ends: very light traffic should leave total delay near mean service time, while approaching saturation from below should send stationary delay upward. Keep one unstable example in the revision set so a negative formula result is recognised as a failed condition rather than accepted as a number.
On the handwritten page, place the stability test beside every M/M/1 expression and add the two reconciliation identities. During timed work, those checks are faster and more reliable than memorising several disconnected formulas.
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